Collect. Math. 44 (1993), 59–70 c 1994 Universitat de Barcelona

Generalized precompactness and mixed topologies

Jurie Conradie

Department of Mathematics, University of Cape Town Rondebosch, 7700 South Africa

Abstract The equicontinuous sets of locally convex generalized inductive limit (or mixed) topologies are characterized as generalized precompact sets. Uniformly pre- Lebesgue and Lebesgue topologies in normed Riesz spaces are investigated and it is shown that order precompactness and mixed topologies can be used to great advantage in the study of these topologies.

Notions of smallness play an important role in analysis, and one of the best known and most useful of these is precompactness. Many smallness properties appearing in the literature can in fact be thought of as “precompactness-like” conditions. A generalized form of precompactness was introduced in [4] in order to prove an ex- tension of Grothendieck’s precompactness lemma. The aim of this paper is to show that in the context of locally convex spaces there is an intimate relationship between generalized precompactness and generalized inductive limit (or mixed) topologies. Generalized precompactness is defined in the first section, and examples are given. Mixed topologies are introduced in the second section and it is shown that the equicontinuous sets of locally convex mixed topologies can often be characterized as generalized precompact sets. In the third section the special case of order pre- compact sets in normed Riesz spaces and the related mixed topologies are explored in more detail. These topologies turn out to be closely related to the uniformly Lebesgue topologies introduced by Nowak in [11].

59 60 Conradie

1. Generalized precompactness

Let E be a (real or complex). A B on E is a collection of subsets of E which covers E, is closed under finite unions and has the property that if B ∈ A ⊂ B, then A ∈B. Now let E be a . Following [4], we call a subset A of E B-precompact if for every neighborhood U of0inE, there is a B ∈Bsuch that A ⊂ B + U. The reader is referred to [4] for the elementary properties of B-precompact sets. Some examples can also be found there; we give some more. Example 1.1: Let E be a and B the collection of relatively weakly compact subset of E. It follows from a result of Grothendieck ([5], Chapter 13, Lemma 2) that the B-precompact sets are exactly the relatively weakly compact sets in E. Example 1.2: Let E be a normed space and B be the collection of bounded weakly metrizable subsets of E. It follows from [14], Lemma 1.2 that the B-precompact sets are exactly the bounded weakly metrizable sets. Example 1.3: Let E be a locally convex space. If U is a closed absolutely convex neighborhood of 0 in E, we shall denote by EU the locally convex space obtained by equipping E with the gauge of U as . Let B be the collection of bounded subsets of E. Then E is quasinormable if for every closed absolutely convex neighbor- hood U of 0 in E, there is another such neighborhood V such that V is B-precompact in EU (cf. [9], 10.5.2). Example 1.4: Let A be an ideal of operators between Banach spaces in the sense of Pietsch ([13]). For a Banach space F let B be the collection of all subsets of B of F such that there is a Banach space G and an S ∈A(G, F ) such that B ⊂ S(BG), where BG denotes the closed unit ball of G. A bounded linear operator T : E → F belongs to the surjective hull of the closure of A if and only if T maps the unit ball of the Banach space E into a B-precompact set. (cf. [10]). A bornology B on a vector space E is a if it is closed under sums, scalar multiples and balanced hulls. A subset B0 of a bornology B is a basis for B if for every B ⊂B, there is a B0 ∈B0 such that B ⊂ B0. A vector bornology will be called convex if it has a basis consisting of absolutely convex sets. It is easy to check that if B is a bornology on a topological vector space E, the collection Bp of all B-precompact sets is again a bornology on E.IfB is a vector bornology, so is Bp;ifB is convex, Bp is closed under the formation of convex hulls. Generalized precompactness and mixed topologies 61

2. Mixed Topologies

The mixed topologies we consider will be a special case of the generalized inductive limit topologies introduced by Garling [7], and a slight generalization of the mixed topologies of Persson [12]. If E is a vector space, B a convex bornology on E and τ a vector topology on E such that every B ∈Bis τ-bounded, we shall call the triple (E,B,τ)amixed space. If in addition B has a basis consisting of τ-closed sets, the mixed space is called normal. The mixed topology γτ (B) is the finest locally convex topology coinciding with τ on the sets in B. If the bornology B is clear from the context, we shall abbreviate γτ (B)toγτ . We write β for the finest locally convex topology for which every B ∈Bis bounded; when equipped with this topology E is a . The space of all linear functionals on E which are bounded on the sets in B will be denoted by Eb. This is also the dual (E,β) of E equipped with the topology β. The space Eb will always have the topology τb of uniform convergence on the sets of B.

It is easy to check that if τ is locally convex, we have τ ≤ γτ ≤ β and hence b b (E,τ) ⊂ (E,γτ ) ⊂ E . Furthermore, (E,γτ ) is a complete subspace of E ([12], Corollary 1.1, Theorem 2.1). In the case where (E,B,τ) is normal, it follows from Grothendieck’s completeness theorem ([15], Chapter VI, Theorem 2) that (E,γτ ) is the closure of (E,τ) in Eb.

It follows from [7], Proposition 1, that if (E,B,τ) is a mixed space, B0 a basis for B and τ locally convex, then a basis for the γτ -neighborhoods of 0 is given by the collection of absolutely convex hulls of the sets ∪{(B ∩ UB): B ∈B0}, where

(UB)B∈B0 ranges over families of absolutely convex τ-neighborhoods of 0 in E. This description enables us to generalize a result of Cooper ([Co], Proposition 1.22) char- acterizing the γτ -equicontinuous sets.

Theorem 2.1

Let (E,B,τ) be a normal mixed space, with τ locally convex and B0 a basis for B consisting of τ-closed sets. A subset A of (E,γτ ) is γτ -equicontinuous if and only if for every ε>0 and every B ∈B0 there is a τ-equicontinuous set A(ε, B) such that A ⊂ A(ε, B)+εB0, where the polar B0 is taken in Eb. 62 Conradie

Proof. We first note the every B ∈B0 is σ(E,(E,τ) )-closed, and hence also σ(E,(E,γτ ) )-closed, by [15], Chapter VI, Theorem 2, Corollary 3. If U is an abso- lutely convex closed τ-neighborhood of 0 in E, it can then be shown as in the proof of [15], Chapter VI, Theorem 2 that (U ∩ B)0 ⊂ U 0 + B0 ⊂ 2(U ∩ B)0.

If A is a γτ -equicontinuous set and ε>0, it follows from the characterization of the γτ -neighborhoods of 0 that we can find a family (UB)B∈B0 of absolutely convex 0 closed τ-neighborhoods of 0 such that A ⊂ ε[ac ∪{(B ∩ UB): B ∈B0}] , (where “ac” denotes the absolutely ) and hence for every B ∈B0, ⊂ ∩ 0 ⊂ 0 0 A ε(B UB) εUB + εB .

0 The result then follows from the fact that εUB is τ-equicontinuous. Conversely, suppose A ⊂ (E,γτ ) satisfies the given condition. It follows eas- ily from the definition of γτ that A will be γτ -equicontinuous if (and only if) the restrictions of the functionals in A to B is τ-equicontinuous for every B ∈B0.If ε>0 and B ∈B0, then it follows from the assumption that we can find a closed ⊂ 1 0 0 ⊂ ∩ 0 absolutely convex τ-neighborhood UB of 0 such that A 2 ε[UB +B ] ε[UB B] . Hence |f(x)|≤ε for every f ∈ A, x ∈ UB ∩ B, as required.  As was pointed out in [4], the fact that for a bornology B on E, ∪B = E,is needed to show that every precompact set is B-precompact. This still holds even if we only assume that ∪B is dense in E. This slightly generalized version of B- precompactness allows us to restate the above theorem.

Corollary 2.2 Let (E,B,τ) be a normal mixed space, E the collection of τ-equicontinuous subsets of (E,τ) and let (E,γτ ) have the topology τb. Then a subset of (E,γτ ) is γτ -equicontinuous if and only if it is E-precompact. It follows from the corollary that B-precompact sets in a duality-setting may well signify the presence of a normal mixed space. We illustrate this using Example 1.1. Let E denote the dual of the Banach space E, B the bornology of bounded sets in E and τ the Mackey topology τ(E,E). Then it is easy to see that (E, B,τ) is a mixed space, and it follows from the fact that the closed unit ball in E is weak∗ compact that it is in fact a normal mixed space. It now follows from Example 1.1 and Theorem 2.1 that the associated mixed topology is in fact the Mackey topology τ(E,E). A further example will be explored in more depth in the next section. Generalized precompactness and mixed topologies 63

3. Order Precompact Sets and Uniformly Lebesgue Topologies

In this section we look at an example of the duality discussed in the previous section in the setting of Riesz spaces. We briefly summarize some of the notions to be used; more information may be found in [1]. All Riesz spaces will be assumed to be Archimedean. If E is a , its set of positive elements will be denoted by E+, the space of all order bounded linear functionals on it by E∼, and the space of order continuous linear functionals by E×. A subset A of E is solid if x ∈ E,y ∈ A and |x|≤|y| implies x ∈ A.Anideal in E is a solid linear subspace; the ideal of E generated by a subset of A of E is denoted by IA.Ifx, y ∈ E, we write [x, y]={z ∈ E: x ≤ z ≤ y} and call such a set an order interval. A set is order bounded if it is contained in an order interval. A linear space topology τ on a Riesz space E is locally solid if it has a basis for the neighborhoods of 0 consisting of solid sets; if it is in addition locally convex, it will be called locally convex-solid. The space of all τ-continuous linear functionals is denoted by (E,τ) (or E for short); if τ is locally solid, (E,τ) is an ideal in E∼. If E is a Banach with dual E, E = E∼. A locally solid topology τ is pre- Lebesgue if every disjoint order-bounded sequence is τ-convergent to 0, Lebesgue if every decreasing net with infimum 0 is τ-convergent to 0, and Fatou if it has a basis for the neighborhoods of 0 consisting of solid order-closed sets. A pre-Lebesgue topology is Lebesgue if and only if it is Fatou. If τ is a Hausdorff Fatou topology, every solid order-closed set is τ-closed. If E is a normed lattice, we shall write Ea × ∩ × (respectively Ea ) for the largest ideal of E (respectively E E ) on which the topology induced by the norm of E is Lebesgue. If E is a locally solid Riesz space and B the bornology of order bounded subsets of E, the B-precompact sets were called Riesz precompact in [2]. These sets are closely related to the order precompact and quasi-order precompact sets of [6]. We recall that a subset A of E is order precompact if for every solid neighborhood U of 0 in E, there is a positive x in the ideal of E generated by A such that A ⊂ [−x, x]+U. We refer to [2] for more detailed information on these notions. Uniformly Lebesgue topologies were introduced by Nowak in [11] in the setting of normed function spaces. In order to generalize this notion to a large class of normed Riesz spaces, we need to generalize the topology of convergence in on sets of finite measure. This is done in [3]; we give the facts pertinent to this paper here. Let E be a Riesz space which contains an order-dense Riesz subspace F which admits a Hausdorff Lebesgue topology τ. The topology τ has a set P of defining 64 Conradie

Riesz pseudonorms. If p ∈P and 0 ≤ u ∈ F , we can define a Riesz pseudonorm pu by pu(x)=p(|x|∧u)(x ∈ E).

The topology τm defined by the pseudonorms pu (p ∈P, 0 ≤ u ∈ F ) is independent of F and τ. It is a Hausdorff Lebesgue topology, and is in fact the coarsest such + topology on E.If(xn) is a disjoint sequence in E , (xn)isτm-convergent to 0. In the case where E is a Riesz space of measurable functions on a semi-finite measure space, τm is the topology of convergence in measure on sets of finite measure. In the rest of this section whenever the existence of the topology τm on a Riesz space E is assumed, it will be assumed that E has an order-dense Riesz subspace which admits a Hausdorff Lebesgue topology. It is known that all Hausdorff Lebesgue topologies induce the same topology on the order bounded subsets of a Riesz space ([1], Theorem 12.9). Using this result it is easy to show that a Hausdorff locally solid topology τ on E is Lebesgue if and only if every order bounded net (xα) which is τm-convergent to 0 is also τ-convergent to 0. This motivates the following definition (see also [11], Definition 1.1).

Let E be a normed Riesz space with unit ball BE. A locally solid topology τ is uniformly Lebesgue if every net in BE which is τm-convergent to 0 is also τ- convergent to 0; and τ is uniformly pre-Lebesgue if every disjoint sequence in BE is τ-convergent to 0. It follows at once that every uniformly (pre-) Lebesgue topology is (pre-) Lebesgue. The converse holds in L∞-spaces. Clearly τm is a uniformly Lebesgue topology. Since disjoint sequences are τm-convergent to 0, every uniformly Lebesgue topology is uniformly pre-Lebesgue. The following duality result will play a crucial role in the rest of this section.

Theorem 3.1 Let E be a normed Riesz space with closed unit ball B and A a solid σ(E∼,E)- ∼ bounded subset of E . Define the seminorm pA on E by pA(x) = sup{|f(x)|: f ∈ A}. Consider the following statements: ∼ (1) B is order pA-precompact and A is order |σ|(E ,E)-precompact. (2) A is order precompact for the norm on E and B is order |σ|(E,IA)-precompact. (3) fn→0 for every disjoint sequence (fn) in A. (4) pA(xn) → 0 for every disjoint sequence (xn) in B. Then (1) ⇒ (4), and if E is a , or if the topology τm can be defined in E, (4) ⇒ (1). If A is a subset of the norm dual E of E, (1) ⇐⇒ (2) ⇐⇒ (3). Generalized precompactness and mixed topologies 65

Proof. This is a special case of [2], Theorem 3.3. If E is a Banach lattice, (4) ⇒ (1) follows from the fact that the norm topology on E is a complete topology for which B is bounded. In the case where the topology τm can be defined on E,itis a Hausdorff Lebesgue topology on E. To see that B is τm-bounded, it suffices to observe that it can be shown (cf. [3], Theorem 5.6) that τm is coarser than the norm topology on E. 

Corollary 3.2 Let E be a normed Riesz space and F an ideal in E. Then the topology induced by the norm of E on F is Lebesgue if and only if the closed unit ball BE of E is order |σ|(E,F)-precompact.

+ Proof. Let BE be order |σ|(E,F)-precompact and f ∈ F . Then (1) ⇒ (3) of 3.1, with A =[−f,f], shows that every disjoint sequence in A is norm convergent to 0, and it follows that the norm of E induces a pre-Lebesgue topology on F . The norm topology is Fatou on E, and since F is an ideal, also on F . It follows that the norm topology on F is Lebesgue. Conversely, if the norm topology on F is Lebesgue, hence pre-Lebesgue, it follows from (3) ⇒ (1) of 3.1 that BE is |σ| (E,F)-precompact. 

Corollary 3.3 Let τ be a locally convex pre-Lebesgue topology on a normed Riesz space E.If

BE is order τ-precompact, τ is uniformly pre-Lebesgue. Conversely, if τ is uniformly Lebesgue, or E is a Banach lattice and τ is uniformly pre-Lebesgue, BE is order τ-precompact.

Proof. Since τ is pre-Lebesgue, every τ-equicontinuous subset A of F =(E,τ) is ∼ order |σ|(F, E)-precompact ([2], Theorem 2.7); also F is an ideal in E .IfBE is order τ-precompact, it follows from (1) ⇒ (4) of 3.1 that τ is uniformly pre-Lebesgue. Conversely, under the stated conditions it follows as before that (4) ⇒ (1) of 3.1 holds, and the result follows. 

Corollary 3.4 If E is a Banach lattice and τ a uniformly pre-Lebesgue locally convex topology ⊂ on E, then F =(E,τ) Ea. 66 Conradie

Proof. This follows from 3.3, 3.2 and the fact that τ is finer than |σ|(E,F). 

The converse of 3.4 does not hold. As an example, let E = L2[0, 1] and τ be the usual norm topology of E. Then (E,τ) = L2[0, 1] = Ea.Ifτ were uniformly pre-Lebesgue, BE would be order τ-precompact (by 3.3), and it would then follow from [8], Lemma 4.4 that L2[0, 1] is finite-dimensional.

Theorem 3.5 Let τ be a locally convex-solid topology on a normed Riesz space E. If every τ- equicontinuous set is an order precompact subset of Ea, τ is uniformly pre-Lebesgue, and the converse holds if E is a Banach lattice.

Proof. If every τ-equicontinuous set A is order precompact in Ea, then in particular ⊂ | | F =(E,τ) Ea. It follows from 3.2 that BE is order σ (E,F)-precompact, and the result then follows from (2) ⇒ (4) of 3.1. Conversely, if E is a Banach lattice, we have F =(E,τ) ⊂ E∼ = E. Also, (4) ⇒ (2) of 3.1 shows that every τ-equicontinuous set A in E is order precompact in | | ⊂  E and that BE is order σ (E,F)-precompact. It follows from 3.2 that F Ea. We immediately obtain a partial converse to 3.4:

Corollary 3.6 | | Let E be a normed Riesz space and F an ideal in Ea. Then σ (E,F) is a uniformly pre-Lebesgue topology. The following result, reminiscent of the fact that Hausdorff Lebesgue topolo- gies coincide on order bounded sets, will be needed to analyse uniformly Lebesgue topologies:

Theorem 3.7

Let τ be a Lebesgue topology on the Riesz space E. Then τm induces a finer topology than τ on the order τ-precompact sets of E.Ifτ is Hausdorff, the two topologies coincide on the order τ-precompact sets.

Proof. Let A be τ-precompact and suppose (xα) is a net in A which is τm-convergent to x ∈ A. It follows from [1], Theorem 12.8 that the topology induced by τm on order intervals of E is finer than that induced by τ. Let p be a τ-continuous pseudonorm ∈ + | |−| |∧ ∈ and ε>0. Choose u IA such that p( x x u) <εfor every x A. Since (xα)isτm-convergent to x, (|x − xa|∧2u)isτm-convergent to 0, and so it follows a Generalized precompactness and mixed topologies 67 priori that we can find an α0 such that p(|x − xa|∧2u) <εfor α ≥ α0. Therefore, for α ≥ α0

p(x − x0) ≤ p(|x − xα|−|x − xα|∧2u)+p(|x − xα|∧2u)

≤ p(|x|−|x|∧u)+p(|xα|∧u)+p(|x − xα|∧2u) < 3ε and so (xα)isτ-convergent to x.Ifτ is Hausdorff, τm ≤ τ on E, and the result follows. 

Proposition 3.8 Let τ be a locally solid topology on a normed Riesz space E.Ifτ is Lebesgue and BE is order τ-precompact, τ is uniformly Lebesgue. The converse holds if τ is locally convex.

Proof. The first part is immediate from Theorem 3.7, and the second follows from Corollary 3.3. 

Corollary 3.9 Let τ be a locally convex uniformly pre-Lebesgue topology on a normed Riesz space E. Then the following are equivalent: (a) τ is uniformly Lebesgue (b) τ is Lebesgue (c) τ is Fatou.

Proof. The implications (a) ⇒ (b) ⇐⇒ (c) are all easy, and (b) ⇒ (a) follows from 3.8 and (4) ⇒ (1) of 3.1, recalling that by convention we are assuming that the topology τm can be defined on E. 

Corollary 3.10 | | × If E is a normed Riesz space, σ (E,Ea ) is a uniformly Lebesgue topology. Proof. Immediate from 3.6 and 3.9.  It is now possible to give a non-trivial example of a uniformly pre-Lebesgue topology which is not uniformly Lebesgue. Let E be a Banach lattice such that × ⊂ ×  | | | | E E = Ea, (E = E ). Then τ = σ (E,E )= σ (E,Ea) is uniformly pre- ⊂ ×  × Lebesgue (by 3.6), but since (E,τ) = Ea E , (Ea = E ),τis not Lebesgue and 68 Conradie therefore not uniformly Lebesgue. An example of such a Banach lattice is the direct sum L∞[0, 1] ⊕ L2[0, 1], equipped with the coordinate-wise ordering and the norm

f ⊕ g = max{f∞, g2}. ∗ The is L∞[0, 1] ⊕ L2[0, 1], with norm

ϕ ⊕ ψ = ϕ∗ + ψ2, ∗ where ·∗ denotes the norm of L∞[0, 1]. Proposition 3.11 Let τ be a uniformly Lebesgue topology on a normed Riesz space E. Then ⊂ × (E,τ) Ea . Proof. Let f ∈ (E,τ) and suppose the sequence (xn) converges in norm to 0. Then there is no loss in generality in assuming that xn ∈ BE for all n ∈ N, and the same argument as in the proof of 3.1 shows that (xn)isτm-convergent to 0. It then follows from 3.3 and 3.7 that (xn)isτ-convergent to 0, and so f(xn) → 0. Hence (E,τ) ⊂ E and the result now follows in the same way as in 3.4. 

Theorem 3.12 A locally convex-solid topology τ on a normed Riesz space E is uniformly Lebesgue if and only if every τ-equicontinuous set is an order-precompact subset × of Ea . × ⊂ Proof. If ever τ-equicontinuous subset is an order precompact subset of Ea Ea, ⊂ × ⊂ × it follows from 3.5 that τ is uniformly pre-Lebesgue; since (E,τ) Ea E ,τ is also Lebesgue and hence by 3.9 uniformly Lebesgue. Conversely, if τ is uniformly ⊂ ∩ × × Lebesgue, (E,τ) Ea E = Ea (by 3.11) and the result then follows from (4) ⇒ (1) of 3.1. 

It is clear from the above result that if the topology τm can be defined on a normed Riesz space E, then there is a finest uniformly Lebesgue locally convex topology on E, namely the topology of uniform convergence on the order precom- × pact subsets of Ea . Likewise it follows from 3.5 that there is a finest uniformly pre-Lebesgue topology on a Banach lattice E, namely, the topology of uniform con- vergence on the order precompact subsets of Ea. We now show that these topologies are often mixed topologies. It is clear from 2.2 that the appropriate mixed spaces B | | × B | | B to consider are (E, , σ (E,Ea )) and (E, , σ (E,Ea)) respectively, where is the bornology of norm-bounded subsets of E.

Lemma 3.13 Let E be a Fatou-normed Riesz space and B its bornology of norm-bounded sets. If τ is a Hausdorff Fatou topology on E such that every B ∈Bis τ-bounded, then (E,B,τ) is a normal mixed space. Generalized precompactness and mixed topologies 69

Proof. Clearly B0 = {nBE: n ∈ N} is a basis for B consisting of solid order-closed sets, and since τ is a Hausdorff Fatou topology, each such set is τ-closed ([1], Theorem 12.7). 

Theorem 3.14 Let E be a Fatou-normed Riesz space, B the bornology of norm-bounded subsets × B | | of E and F an ideal in Ea which separates the points of E. Then (E, , σ (E,F)) is a normal mixed space and γ|σ|(E,F) is the topology of uniform convergence on the × order precompact sets of the norm closure F of F in E . In particular, if Ea separates the points of E, the finest uniformly Lebesgue topology is a mixed topology.

Proof. It is enough to note that |σ|(E,F) is a Hausdorff Lebesgue, hence Fatou, topology with dual F . 

If the topology τm can be defined on a Fatou-normed Riesz space E, 3.13 shows that (E,B,τm) is also a normal mixed space. Since τm is in general not a locally convex topology, Theorem 2.1 cannot be used to identify the mixed topology γτm . However, Theorem 3.7 comes to the rescue in many cases.

Lemma 3.15 × Let E be a normed Riesz space such that Ea separates the points of E. Then | | × τm and σ (E,Ea ) coincide on the closed unit ball BE of E. | | × Proof. The topology σ (E,Ea ) is Hausdorff and hence τm can be defined on E. The result now follows from 3.2 and 3.7. 

To formulate the next result, we introduce the notation γτ for the finest vector topology which coincides with a vector topology τ on the norm-bounded subsets of a normed Riesz space E. It follows from [7], Proposition 5 that if τ is locally convex γτ = γτ . Theorem 3.16 × Let E be a normed Riesz space such that Ea separates the points of E. Then × × γτm = γ| | = γ = γτ . σ (E,Ea ) |σ|(E,Ea ) m

Proof. The result follows at once from 3.15 and the above remark. 

Corollary 3.17 (cf. [11], Theorem 3.3) Let E be a normed Riesz space. The following are equivalent: (a) γτ is locally convex ×m (b) Ea separates the points of E | | × (c) τm and σ (E,Ea ) coincide on the closed unit ball BE of E. 70 Conradie

Proof. (a) ⇒ (b): Since γ is clearly a uniformly Lebesgue topology, (E,γ ) ⊂ E× τm τm a by 3.11. If γ is locally convex, it follows that E× separates the points of E. τm a (b) ⇒ (c) : This is 3.15. (c) ⇒ (a) : This follows as in 3.16. 

Proposition 3.18 Let E be a normed Riesz space. Then γ is the finest uniformly Lebesgue τm topology on E.

Proof. Clearly γ is uniformly Lebesgue. If τ is a uniformly Lebesgue topology on τm E,τ is finer than τ on B , and so τ ≤ γ ≤ γ .  m E τ τm Applications of the material in this section to the characterization of compact sets and compact operators in Banach lattices will be given in a forthcoming paper.

References

1. C.D. Aliprantis and O. Burkinshaw, Locally solid Riesz spaces, Academic Press, New York, 1978. 2. J.J. Conradie, Duality results for order precompact sets in locally solid Riesz spaces. Indag. Math. N.S. 2 (1991), 19–28. 3. J.J. Conradie, The coarsest Hausdorff Lebesgue topology, preprint. 4. J.J. Conradie and J. Swart, A general duality result for precompact sets, Indag. Math. N.S. 1 (1990), 409–416. 5. J. Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics 92, Springer- Verlag, New York-Berlin-Heidelberg-Tokyo, (1984). 6. M. Duhoux, Order precompactness in topological Riesz spaces, J. London Math. Soc. (2) 23 (1981), 509–522. 7. D.J.H. Garling, A generalized form of inductive limit topology for vector spaces, Proc. Lond. Math. Soc. 14 (1964), 1–28. 8. J.J. Grobler, Indices for Banach function spaces, Math. Z. 145 (1975), 99–109. 9. H. Jarchow, Locally convex spaces, Teubner, Stuttgart, 1981. 10. H. Jarchow and U. Matter, Interpolative constructions for operator ideals, Note di Matematica 8 (1988), 45–56. 11. M. Nowak, Mixed topology on normed function spaces, I, Bull. Pol. Ac. Math. 36 (1988), 251–262. 12. A. Persson, A generalization of two-norm spaces, Ark. Mat. 5 (1963), 27–36. 13. A. Pietsch, Operator Ideals, North Holland, Amsterdam, 1980. 14. N. Robertson, Asplund operators and holomorphic maps, Manuscripta Math. 75 (1992), 25–34. 15. A.P.Robertson and W. Robertson, Topological VectorSpaces, Cambridge University Press, 1966.